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Frobenius splitting and geometry of \(G\)-Schubert varieties. (English) Zbl 1160.14035
Let \(G\) be a connected reductive group over an algebraically closed field \(k\) of positive characteristic and \(B\) be a Borel subgroup of \(G\). Let \(X\) be an equivariant embedding of the group \(G\). A \(G\)-Schubert variety in \(X\) is a subvariety of the form \(\text{diag}(G)\cdot V\), where \(V\) is a \(B\times B\)-orbit closure in \(X\). If \(X\) is the wonderful compactification of a semisimple group of adjoint type, then \(G\)-Schubert varieties coincide with the closures of G. Lusztig’s \(G\)-stable pieces [Parabolic character sheaves. I. Mosc. Math. J. 4, No. 1, 153–179 (2004; Zbl 1102.20030); II. Mosc. Math. J. 4, No. 4, 869–896 (2004; Zbl 1103.20041)].
The authors prove that \(X\) admits a Frobenius splitting which is compatible with all \(G\)-Schubert varieties. Moreover, when \(X\) is smooth, projective and toroidal, then any \(G\)-Schubert variety in \(X\) admits a stable Frobenius splitting along an ample divisor. On the other hand, an example of a nonnormal \(G\)-Schubert variety in the wonderful compactification of a group of type \(G_2\) is given. In the last section, a generalization of the Frobenius splitting results to the more general class of \(R\)-Schubert varieties is obtained.

14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text: DOI arXiv
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